Methods and systems for optical beam steering

ABSTRACT

An integrated optical beam steering device includes a planar dielectric lens that collimates beams from different inputs in different directions within the lens plane. It also includes an output coupler, such as a grating or photonic crystal, that guides the collimated beams in different directions out of the lens plane. A switch matrix controls which input port is illuminated and hence the in-plane propagation direction of the collimated beam. And a tunable light source changes the wavelength to control the angle at which the collimated beam leaves the plane of the substrate. The device is very efficient, in part because the input port (and thus in-plane propagation direction) can be changed by actuating only log 2  N of the N switches in the switch matrix. It can also be much simpler, smaller, and cheaper because it needs fewer control lines than a conventional optical phased array with the same resolution.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. application Ser.No. 16/284,161, now U.S. Pat. No. 10,649,306, filed Feb. 25, 2019, andentitled “Methods and Systems for Optical Beam Steering,” which is acontinuation application of U.S. application Ser. No. 15/630,235, nowU.S. Pat. No. 10,261,389, filed Jun. 22, 2017, and entitled “Methods andSystems for Optical Beam Steering,” which in turn claims the prioritybenefit, under 35 U.S.C. § 119, of U.S. Application No. 62/353,136,filed Jun. 22, 2016, and entitled “Integrated Lens-Enabled LIDARSystem.” Each of these applications is incorporated by reference herein.

GOVERNMENT SUPPORT

This invention was made with Government support under Contract No.FA8721-05-C-0002 awarded by the U.S. Air Force. The Government hascertain rights in the invention.

BACKGROUND

The meteoritic rise of autonomous navigation in real-world settings forself-driving cars and drones has propelled rapidly growing academic andcommercial interest in LIDAR. One of the key application spaces that hasyet to be filled, but is of great interest, is a non-mechanicallysteered LIDAR sensor which has substantial range (e.g., 100-300 m), lowpower (e.g., 1-10 W), low cost (e.g., hundreds of dollars), highresolution (e.g., 10⁴ to 10⁶ pixels) and small size (e.g., 10 cm³).There are several candidate technologies including micro-mechanicalmirrors, liquid-crystal based devices, and integrated photonics that arecurrently being explored academically and commercially to fill thisniche.

Current state-of-the-art chip-scale integrated-photonic LIDARs are basedon 1D or 2D phased array antennas. In this type of architecture, a 1D or2D array of dielectric grating antennas is connected toelectrically-controlled thermo-optic (TO) or electro-optic phaseshifters. These phase shifters are fed by waveguides splitting off fromone main dielectric waveguide which brings power from an off-chip oron-chip source. By applying a gradient to the phases tuning eachantenna, in-plane or out-of-plane beam-steering can be enabled.

The direct predecessor of this architecture are radio frequency (RF)phased arrays developed for military and commercial RADARs. Although thedetailed implementation is different because RF primarily relies onmetallic waveguides and structures whereas integrated photonics usesdielectrics, optical phased arrays are essentially based on directlyreplacing RF components with their optical equivalents. This directtranslation brings a significant disadvantage: whereas metallicwaveguides can be spaced at sub-wavelength pitches, eliminatingparasitic grating lobes, dielectric waveguides have to be separated byseveral wavelengths to prevent excessive coupling, resulting insignificant grating lobes.

RF phased array radars are routinely produced with closely spacedantennas (<λ/2 apart) in subarrays that can be tiled to create verylarge apertures. This provides wide-angle steering and scaling to largepower-aperture designs. Fundamentally, the radiating elements can beclosely spaced with independent control circuitry because theamplifiers, phase shifters and switches in the RF are implemented assubwavelength lumped elements.

Current chip-scale optical phased arrays often reproduce RF phased arrayarchitectures, with RF elements replaced with their optical analogs.Fundamentally, the optical analogs to RF components are traveling-wavedesigns that are multiple wavelengths long and spaced apart by more thanλ/2. This design allows beam-steering over very small angles. In anend-fed geometry, for example, the grating antenna elements can beclosely spaced for wide-angle azimuthal steering and use wavelengthtuning to change the out-coupling angle of the gratings for elevationsteering. But this end-fed geometry cannot be tiled without introducingsignificant grating lobes due to its sparsity.

SUMMARY

Embodiments of the present technology include an optical beam steeringapparatus comprising a substrate, a plurality of waveguides formed onthe substrate, a planar dielectric lens formed on the substrate inoptical communication with the waveguides, and an output coupler formedon the substrate in optical communication with the planar dielectriclens. In operation, the waveguides comprise a first waveguide and asecond waveguide. The planar dielectric lens collimates light emitted bythe first waveguide as a first collimated beam propagating in a firstdirection in a plane of the substrate and collimates light emitted bythe second waveguide as a second collimated beam propagating in a seconddirection in the plane of the substrate different than the firstdirection. And the output coupler guides the first collimated beam inthe first direction and the second collimated beam in the seconddirection and couples at least a portion of the first collimated beamand the second collimated beam out of the plane of the substrate.

The optical beam steering apparatus may include at least 32, 100, or1000 waveguides in optical communication with the planar dielectriclens. The light emitted by these waveguides may not be phase coherent(e.g., the first waveguide may have an arbitrary phase relative to thelight emitted by the second waveguide).

The planar dielectric lens may have a shape selected to satisfy the Abbésine condition. It can have a single focal point or multiple (i.e., twoor more) focal points.

The output coupler can comprise a one-dimensional grating configured todiffract the first collimated beam and the second collimated beam out ofthe plane of the substrate. It could also include a two-dimensionalphotonic crystal that couples the first and second collimated beams outof the substrate.

Examples of such an optical beam steering apparatus may also include atunable light source in optical communication with the waveguides. Thistunable light source tunes a wavelength of the light emitted by thefirst waveguide and the light emitted by the second waveguide. Forinstance, the tunable light source may tune the wavelength to steer thefirst collimated beam to one of at least 15, 50, 100, or 1000 resolvableangles with respect to a surface normal of the plane of the substrate.

These examples may also include a network of optical switches formed onthe substrate in optical communication with the tunable light source andthe waveguides. This network guides the light emitted by the firstwaveguide from the tunable light source to the first waveguide when in afirst state and guides the light emitted by the second waveguide fromthe tunable light source to the second waveguide when in a second state.In cases where there are N waveguides, switching from the first state tothe second state involves actuating up to log₂ N optical switches in thenetwork of optical switches. The optical beam steering apparatus mayalso include a plurality of optical amplifiers formed on the substratein optical communication with the network of optical switches and thewaveguides. These amplifiers amplify the light emitted by the firstwaveguide and the light emitted by the second waveguide.

Other examples of the present technology include a lidar with a lightsource, a network of optical switches, a planar dielectric lens, and aperiodic structure. In operation, the light source emits a beam oflight. The network of optical switches, which are in opticalcommunication with the tunable light source, guides the beam of light toa first waveguide in a plurality of waveguides. The planar dielectriclens, which is in optical communication with the waveguides and has ashape selected to satisfy the Abbé sine condition, collimates the beamof light emitted by the first waveguide as a first collimated beampropagating in a first direction. And the periodic structure, which isin optical communication with the planar dielectric lens, diffracts atleast a portion of the first collimated beam at an angle with respect tothe first direction.

All combinations of the foregoing concepts and additional conceptsdiscussed in greater detail below (provided such concepts are notmutually inconsistent) are part of the inventive subject matterdisclosed herein. In particular, all combinations of claimed subjectmatter appearing at the end of this disclosure are part of the inventivesubject matter disclosed herein. The terminology used herein that alsomay appear in any disclosure incorporated by reference should beaccorded a meaning most consistent with the particular conceptsdisclosed herein.

BRIEF DESCRIPTIONS OF THE DRAWINGS

The skilled artisan will understand that the drawings primarily are forillustrative purposes and are not intended to limit the scope of theinventive subject matter described herein. The drawings are notnecessarily to scale; in some instances, various aspects of theinventive subject matter disclosed herein may be shown exaggerated orenlarged in the drawings to facilitate an understanding of differentfeatures. In the drawings, like reference characters generally refer tolike features (e.g., functionally similar and/or structurally similarelements).

FIG. 1A shows an optical beam-forming device with a planar dielectriclenses.

FIG. 1B shows a bifocal lens suitable for use in an optical beam-formingdevice like the one shown in FIG. 1A.

FIG. 1C shows a bootlace lens suitable for use in an opticalbeam-forming device like the one shown in FIG. 1A.

FIGS. 2A-2C show illustrate steering an optical beam with the opticalbeam-forming device of FIG. 1A.

FIGS. 3A and 3B illustrate transmitting and receiving, respectively,with the optical beam-forming device of FIG. 1A.

FIG. 4A shows a simulated aperture pattern of an optical beam-formingdevice with a planar dielectric lens.

FIG. 4B shows the simulated far-field directivity and far-field beamangles for the ideal aperture given in FIG. 4A

FIG. 5A shows ray-tracing simulations used to determine the optimal portposition and relative angle.

FIG. 5B is a heat plot showing far-field beam spots in u_(x) and u_(y)space.

FIG. 5C shows three-dimensional (3D) beam patterns corresponding tothose in FIG. 5B.

FIG. 6A illustrates beam steering with a conventional optical phasedarray using phase shifters.

FIG. 6B illustrates beam steering with a planar dielectric lens andswitch matrix.

FIG. 7 is a block diagram of an integrated LIDAR chip with a planardielectric lens for optical beam steering.

FIG. 8 shows a LIDAR with a tunable on-chip source that is modulated bya microwave chirp.

FIG. 9 shows an integrated optical beamforming system that scales thebasic design of FIG. 8 to add functionality for N independentlycontrollable beams.

FIG. 10A shows how the unit cells shown in FIGS. 8 and 9 can be modifiedfor tiling an M by N array.

FIG. 10B shows the tiling of the unit cells to form larger apertures.

FIG. 11 shows 3 dB overlapped far-field beam patterns for a lens-enabledbeam-forming system.

FIG. 12A shows a simulation of a far-field beam pattern to extract phasecenter.

FIG. 12B shows a simulation of a far-field beam pattern to extractgaussian beamwidth.

FIG. 12C shows a 2D simulation of on-axis port excitation of a lensfeed.

FIG. 12D shows a 2D simulation of off-axis port excitation of the lensfeed.

DETAILED DESCRIPTION

Although the optical analogy to RF phased arrays has been well explored,there is an entire class of planar-lens based devices developed in theRADAR literature that performs the same function as an RF phased arraywith integrated-photonics analogs. Instead of relying upon manycontinuously tuned thermal phase shifters to steer beams, anintegrated-photonics device excites the focal plane of a speciallydesigned planar lenses to generate a discrete far-field beam.

This approach to making an on-chip beam-steering device (e.g., for LIDARapplications) starts with widely-spaced transmit/receive waveguides(>10λ, apart) that include SOAs, phase shifters, directional couplers,and RF photodiodes. The beam-steering device uses a wide-angle planardielectric lens—an optical equivalent of a Rotman lens for RFbeamforming—to convert the sparse array of waveguides into a dense arrayof output waveguides (˜λ/2 apart) to enable wide-angle steering.Exciting a given input port to the planar dielectric lens steers thebeam in the plane of the lens, and changing the beam's wavelength steersthe beam out of the plane.

This device can be tiled with optical overlapped subarrays to suppresssidelobes. These passive beamforming structures can be realized usingsilicon technology and may be butt-coupled to active photonic chips thatprovide the active transmit/receive functions. Fixed phase shifts in thebeamformer chip could create a sinc-like pattern in the near-field inthe vertical direction. This transforms into a rectangular beam patternin the far-field, suppressing the sidelobes. Advantages of using passivebeamformers with overlapped subarrays include: (1) a dramatic reductionin the number of control lines needed; and (2) much reduced electricalpower dissipation per chip.

Optical Beam-Steering Device Architectures

FIG. 1A shows an example lens-enabled integrated photonic LIDAR system100 with light propagating through each component of the system 100,which is formed on a substrate 104. Light is coupled into the systemfrom a fiber 90 via an on-chip waveguide 102. The waveguide 102 isformed by a 200 nm thick and 1 μm wide Silicon Nitride (SiN) sectionencapsulated in SiO₂ (an upper SiO₂ layer is omitted from the diagram).This fiber 90 connects to a tunable IR source 80 that emits lightcentered at a wavelength of about 1550 nm. The light source 80 can bemated or bonded to the substrate 102 or integrated onto the substrate102. There may also be a preamplifier coupled between the light source80 and the fiber 90 or disposed on the substrate 104 in opticalcommunication with the waveguide 102 to amplify the input beam from thelight source 80.

The waveguide 102 guides the light from the IR source 80 to a switchmatrix 110 composed of Mach-Zehnder interferometers (MZIs) 112. The MZIs112 can switch the input into any one of 2^(D) output ports 114, where Dis the depth of the tree in the switch matrix 110. The optical pathlength of at least one arm of each MZI 112 is controlled by anintegrated thermo-optic (TO) phase shifter (not shown; externalelectronic control lines are also omitted), which allows the opticalbeam to be electronically switched between two output ports 114.

The system 100 may also include semiconductor optical amplifiers (SOAs;not shown) integrated on the substrate 104 before or after the switchmatrix 110. These SOAs can be turned on and off, depending on whether ornot light is propagating through them, to reduce power consumption.

Each output from the switch matrix 110 feeds into a slab waveguide 122that is patterned to form the focal surface of a wide-angle planardielectric lens 120, also called an aplanatic lens. This lens may obeythe Abbé Sine condition and has a lens shape designed for wide-anglesteering. There are many different shapes and structures that providewide-angle steering, including a basic parabolic shape for the lens.Other lens shapes are also possible, including those for a bifocal lens(shown in FIG. 1B), bootlace lens (shown in FIG. 1C), integratedLuneburg lens, Rotman lens, or standard compound lens, such as anachromatic doublet.

The light 121 rapidly diffracts upon entering the slab waveguide 122until being collimated by the aplanatic lens 120. The lens 120 is formedby a patterned PolySi layer 20 nm or 40 nm thick. The collimated beam121 propagates into a output coupler, implemented here as aone-dimensional (1D) grating 130 that guides the collimated beam 121within the plane of the substrate and scatters the collimated beam outof the plane of the substrate 104 and into free-space. The grating 130is formed out of a PolySi layer the same height as the aplanatic lens120.

In other examples, the lens redirects the beam without collimating it toa curved grating (instead of the straight grating 130 shown in FIG. 1A).The curved grating's curvature is selected to collimate the beamredirected by the lens. In other words, the lens and curved grating worktogether to produce a collimated beam.

The MZI switch matrix 120 can be replaced by a 3 dB splitting tree (notshown) that illuminates all of the input ports 114 in parallel. In thisimplementation, beam-steering of this “fan-beam” is accomplished throughfrequency tuning. A separate aperture with an array of detectorsprocesses the LIDAR return. At 100 m, a raster scan for a megapixelscale sensor would take 1 second, based on time-of-flight—far too slowfor most applications—whereas parallelization can allow an acquisitionon the scale of milliseconds. The tradeoff is that the powerrequirements are increased by a factor of N and more detector hardwareis required.

In other embodiments, the output coupler is implemented as atwo-dimensional (2D) photonic crystal instead of a 1D grating. Forinstance, photonic crystal resonances based on bound states in thecontinuum (BIC) can implement the out-of-plane steering. BICs areinfinite quality factor resonances arising due to interference effects.For a broad range of neighboring wave vector points, the quality factoris also very high, potentially enabling large scale photonic structureswith efficient and focused emission across a wide range of directions.

One factor enabling improved performance of a 2D photonic crystalrelative to other output couplers is the rigorously optimal radiationquality factor of BICs. BIC photonic crystal gratings can have qualityfactors as high as 10⁵, enabling propagation over the order of 10⁵periods, or 10 cm, before significantly attenuating. These can easily beused to not just make 50% illuminated, 1 mm long gratings, but even 70%and 95% illuminated, 1 cm long gratings. Slight tuning of the structurecan change the quality factor and allow the output to be tapered for amore even aperture power distribution and a smaller beamwidth in the farfield.

The BICs close to k=0 points, protected by symmetry mismatch between theelectromagnetic resonance and radiation continuum, feature manyadditional desirable properties. The relatively flat and homogeneousbands here allow rapid tuning rates of the emission direction byslightly tuning the frequency of excitation. The angular tuning rate canbe estimated as a function of frequency using the following simplerelation dθ=dω/ω*n_(group), where n_(group) is the group index of theband. For n_(group)=3, this yields a standard tuning rate of 0.1degrees/nm at 1500 nm. Using a flat band with n_(group) as high as 10,the expected tuning rates are about 0.4 degrees/nm, allowing a potentialreduction in the required bandwidth and tuning range of on-chip devicesby a factor of 2 or more.

In addition, the radiation field can also be designed to be highlyasymmetric by breaking the top-down mirror symmetry of the structure, sothat light can be more efficiently collected into the desired direction.This enables lower losses and fewer interference effects from radiationinto the substrate.

Finally, a common problem with arrays of dielectric grating antennas inconventional optical beam-steering devices is their sensitivity tocoupling at wavelength scale pitches. 2D photonic crystals do notexhibit this problem because they are designed for the strong “coupling”regime: the entire structure is wavelength scale.

This reduced sensitivity comes at a price. The in-plane steering andout-of-plane steering are no longer separately controlled by on-chipbeamforming or frequency-tuning: there is a mixture. Despite this, it ispossible to map a given set of frequency and port settings to a givenset of beam angles with a lookup table.

Non-Mechanical Optical Beam Steering with a Planar Dielectric Lens

FIGS. 2A-2C illustrate non-mechanical beam steering for the chip 100shown in FIG. 1A. Non-mechanical beam steering is implemented by twomechanisms. The first is port switching, which changes the in-planepropagation direction of the beam as shown in FIGS. 2A and 2B. Thedesired input port, and hence the desired in-plane propagationdirection, can be selected by setting the MZIs 112 in the opticalswitching matrix 110 to route the input beam. Thus, there may be up toone resolvable in-plane angle for each input port to the planardielectric lens. Systems with 32, 100, 1000, or 10,000 input ports wouldhave up to 32, 100, 1000, or 10,000 resolvable in-plane steering angles,respectively.

As depicted in FIGS. 2B and 2C, the wavelength roughly controls theout-of-plane angle, that is, the angle between the beam center and thez-axis. Thus, there may be up to one resolvable out-of-plane angle foreach wavelength resolvable by the output coupler. For example, systemswith gratings that can resolve 15, 50, 100, or 1000 angles would have upto 15, 50, 100, or 1000 resolvable out-of-plane steering angles,respectively.

These 2D beam steering mechanisms are similar to those of RF Rotmanlenses feeding arrays of patch antennas. The 3D directivity patterns ofthe generated beams are depicted in each subfigure. The precisemathematical relationship between the emission angles and the analyticform of the directivity pattern are detailed below.

FIGS. 3A and 3B show how the system can be used to transmit and receive,respectively. Transmission works as described above: exciting an inputto the planar dielectric lens yields a plane wave that propagates in agiven direction within the plane of the lens, and tuning the wavelengthchanges the propagation angle with respect to the surface normal of theplane of the lens. Receiving works in reverse: the grating collectsincident light, and the lens focuses the incident light on the inputport associated with the corresponding in-plane angle-of-arrival. Theout-of-plane angle of arrival corresponds to the angle with thestrongest transmission, which represents a direct “reflection” from theobject being interrogated. The coupler may be illuminated by light fromother angles, e.g., caused by scattering or indirect “reflections,” butthis light generally is not properly phased-matched to the grating andtherefore does not efficiently couple into the grating.

To better understand the system's operation, consider an ideallypreforming aperture. In operation, an ideal implementation of the planardielectric lens generates a plane wave propagating at a finite angle.The scattered light from the plane wave propagation through the 1Dgrating forms the near-field of the radiation pattern. Assuming that theplane wave emitted from the lens is uniform, that the lens introducesnegligible aberrations, and that the lens and grating parameters arewavelength and angle independent yields a simplified aperture pattern ofthe following form:

${A\left( {x,y} \right)} = \left\{ \begin{matrix}{{\exp\left( {- {qx}} \right)}{\exp\left( {{ik}_{0}u_{x\; 0}x} \right)}{\exp\left( {{ik}_{0}u_{y\; 0}y} \right)}} & \begin{matrix}{{0 \leq x \leq L},} \\{{{- \frac{W}{2}} + {x\;{\tan\left( \phi^{\prime} \right)}}} \leq y \leq {\frac{W}{2}x\;{\tan\left( \phi^{\prime} \right)}}}\end{matrix} \\0 & {else}\end{matrix} \right.$

FIG. 4A shows this ideal near-field aperture pattern. The pattern inFIG. 4A can be thought of as a parallelogram with uniform amplitude inthe y-direction, and exponentially decaying amplitude in thex-direction, determined by the grating decay parameter q where L is thelength of the grating and W is the width. The inclination of theparallelogram is determined by the grating propagation angle ϕ′, whichis derived below and is close in magnitude to the propagation angle ofthe beam output from the lens ϕ_(in)

u_(x,0)=sin(ϕ₀)cos(θ₀) and u_(y,0)=sin(ϕ₀)sin(θ₀) characterize thedirection of the emitted mode and can be calculated by tracking thephase accumulated by the collimated rays emitted from the lens anddiscretely sampling them at the grating teeth. As shown below, these canwrite these as:

$\begin{matrix}{{u_{\gamma,0} = {n_{1}{\sin\left( \phi_{in} \right)}}}{u_{x,0} = {\frac{k_{x,{avg}}\left( \phi_{in} \right)}{k_{0}} - \frac{\frac{2\pi\; m}{\Lambda}}{k_{0}}}}} & (1)\end{matrix}$where k_(x,avg) is the average k component in the grating, n₁ is theeffective index of the TE slab mode in the lens, m is the grating order,and Λ is the grating period. The function of the grating can beunderstood from this equation: it allows phase matching to radiatingmodes through the addition of the crystal momentum 2πm/Λ.

Making the approximation k_(avg)≈n_(eff)k₀ cos(ϕ_(in)), where n_(eff) isthe average effective index of the gratings, makes it possible to showthat u_(x0) and u_(y0) satisfy an elliptical equation:

$\begin{matrix}{{\left\lbrack \frac{u_{x,0} + \frac{m\;\lambda}{\Lambda}}{n_{eff}} \right\rbrack^{2} + \left\lbrack \frac{u_{y,0}}{n_{1}} \right\rbrack^{2}} = 1} & (2)\end{matrix}$This elliptical equation has a simple physical interpretation. Switchingports in-plane takes us along an elliptical arc in u_(x,0) and u_(y,0)space, while tuning the wavelength λ tunes this arc forward andbackwards as depicted in FIG. 6B (described below).

Analytically calculating the directivity, which characterizes thefar-field distribution of radiation, yields:

$\begin{matrix}{{{D\left( {{\Delta\; u_{x}},{\Delta\; u_{y}}} \right)} = {\frac{{Wk}_{0}^{2}{\cos\left( \theta_{0} \right)}}{\pi\;{q\left( {1 - {\exp\left( {{- 2}qL} \right)}} \right)}}\frac{\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y}} \right)}}{1 + {\frac{k_{0}^{2}}{q^{2}}\left( {{\Delta\; u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)^{2}}} \times \left( {1 - {2{\cos\left( {k_{0}{L\left( {{\Delta\; u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)}} \right)}{\exp\left( {- {qL}} \right)}} + {\exp\left( {{- 2}qL} \right)}} \right)}}\mspace{20mu}{{{where}\mspace{14mu}\Delta\; u_{x}} = {{u_{x} - {u_{x,0}\mspace{14mu}{and}\mspace{14mu}\Delta\; u_{y}}} = {u_{y} - {u_{y,0}.}}}}} & (3)\end{matrix}$

FIG. 4B is a plot of the computed far-field directivity and far-fieldbeam angles given in Eq. (3) for the ideal aperture given in FIG. 4A.The peaks to the left and right of the main beam are known as sidelobesand originate from the sinc factor in the directivity function. Thereare several recognizable features to this function, such as the sincfrom the rectangular aperture and the 1/(k₀ ²+q²) from the exponentialdecay. The tan 0′ components introduce a “shear” into the beam spots andcome directly from the tilted aperture pattern.

This result can be used to calculate estimates for the number ofresolvable points for port switching and wavelength steering.Specifically, for wavelength tuning, the result is:

$\begin{matrix}{N_{wavelength} \approx {{{Qn}_{eff}\frac{\Delta\;\lambda}{\lambda_{0}}\frac{v_{g}}{c}} + 1}} & (4)\end{matrix}$where Q is the quality factor of the grating, v_(g) is the groupvelocity of propagation in the grating, and Δλ is the bandwidth. Thisexpression exactly resembles what would be extracted from otherphase-shifter based architectures which rely on frequency tuning forbeam-steering in one direction. The number of resolvable points forsteering in plane is approximately:

$\begin{matrix}{{N_{{in}\text{-}{plane}} \approx \frac{2D_{peak}}{3}} = \frac{2\pi\; W}{3\lambda}} & (5)\end{matrix}$Planar Dielectric Lens Design

The wide-angle planar dielectric lens has a shape selected to satisfythe Abbé Sine condition, which eliminates the Coma aberration. Lensesdesigned this way tend to have good off-angle performance to ±20° or30°. In practice, this quantity can translate to a field of view of 80°or more in ϕ₀. The lens design depends on the focal length, lensthickness, and lens index (the ratio of effective indices of atransverse electric (TE) mode in a SiN slab (n₂) to a SiN slab with alayer of PolySi (n₁)). After creating the lens, the focal plane can beidentified by conducting ray-tracing through the lens and optimizing thefeed position and angle based on maximizing the 2D directivity from the1D aperture pattern computed from ray-tracing.

Following this, ray tracing is done through the grating to compute thefull 3D directivity for several optimized port locations and angles. Theaperture pattern can be extracted from ray-tracing through the gratingin the following way:

$\begin{matrix}{{A\left( {x,y} \right)} = \left\lbrack {\sum\limits_{n = 0}^{N_{rays}}{\sum\limits_{m = 0}^{N_{grat}}{\sqrt{P_{n,m}}{\delta\left( {x - {m\;\Lambda}} \right)}{\delta\left( {y - y_{n,m}} \right)}{\exp\left( {- {qx}} \right)}{\exp\left( {i\;\phi_{n,m}} \right)}}}} \right\rbrack} & (6)\end{matrix}$where the ray amplitudes P_(n,m) and accumulated ray phases ϕ_(n,m) arediscretely sampled for all N_(ray) by N_(grat), ray-gratingintersections at [x_(n,m), y_(n,m)]. The physical interpretation of thisis that each ray-grating intersection acts as a point radiation sourcedriven by the traveling wave (see below). An artificially “added”amplitude decay of exp(−qx) accounts for the grating radiation as therays propagate. The power associated with a given ray-gratingintersection P_(n,m) is calculated from the feed power based onconservation arguments: P_(feed)(ϕ)dϕ=P_(n,m)(y)dy_(n,m)

FIGS. 5A-5C illustrate a full ray-tracing calculation and a 2D aperturepattern extracted by this method for lens-enabled chip-scale LIDARgenerated with λ₀=1.55 μm, q=0.025 μm⁻¹, Λ=700 nm, duty cycle=0.1, feedbeamwidth of 15°, and effective indices n₁=1.39 and n₂=1.96. In FIG. 5A,ray-tracing simulations are used to determine the optimal port positionand relative angle. These rays are traced through the grating and forman aperture pattern. The Fourier transform of the aperture gives theFar-field pattern. Numerical details of effective index calculations,port phase center, and feed patterns are detailed below.

FIG. 5B is a heat plot showing far-field beam spots in u_(x) and u_(y)space. The location of these ports is governed by the equations above,where the beams along the elliptical curve are generated from portswitching, while the points formed from translating the ellipse to theright and left correspond to frequency tuning over ±50 nm around λ₀=1.55μm. FIG. 5C shows three-dimensional (3D) beam patterns corresponding tothose in FIG. 5B. The different shadings indicate the differentwavelengths used to generate the beam. The drooping effect of the beamsas they turn off-axis is caused by increasing in-plane momentum.

Ray-tracing through the grating is valid in the regime where the gratingteeth individually cause low radiation loss and small incoherentreflections (i.e., the excitation frequency is far from the Braggbandgap). When correct, this method is useful because it can be used tocompute the aperture pattern quickly for a large many-wavelengthstructure while including the effects of lens aberrations and anonuniform power distribution, two features which would be difficult tomodel analytically, and very costly to simulate through 2D or 3Dfinite-difference time-domain (FDTD) techniques.

Following standard RADAR design procedures, once a set of far-field 3Ddirectivity patterns are calculated, new ports are placed to overlap thegain at each port by 3 dB to provide suitable coverage in the field ofview. To confirm the successful operation of this design, the aperturepatterns for multiple wavelengths between 1500 and 1600 nm arecalculated for all ports. The wavelength dependence of the effectiveindices and the grating decay factor q are included. Directivitypatterns are plotted in u_(x),u_(y) space for a range of wavelengths inFIG. 5B, where the 3 dB overlapped ports lie along an elliptical arc,and where the arc is translated forward and backward by tuning thewavelength. These same beams are plotted in 3D in FIG. 5C. Note thatbeams towards the edge of the field of view tend to “fall-into” thedevice plane because of increasing in-plane momentum (see below).

Ray-tracing is one method that can be used to design an opticalbeam-steering chip with a planar dielectric lens. The parameters usedfor this method, such as the port phase centers, feed beam width,grating decay length, and the effective indices, can be extracted fromother calculations. In addition, many other simulations may beundertaken to validate the assumptions of our ray-tracing computationsto account for second order effects. Finally, the outcome of theray-tracing calculations may be compared to the analytically predicteddirectivity functions and beam directions to assess the performance andvalidity. Once a design is validated, cadence layouts of the necessarycomponents can be generated automatically and verified to ensure theysatisfy design rule checks based on fabrication limitations and otherphysics-based constraints.

Performance of Optical Beam Steering with a Planar Dielectric Lens

The optical beam steering architecture shown in FIG. 1A has severaladvantages over phase-shifter based solutions. RF lenses were developedin part to reduce or minimize the use of phase shifters, which areexpensive, lossy, complicated, power hungry and bulky. Some of the sameconsiderations apply here: thermo-optic phase shifters are power hungrycomponents, typically using 10 mW or more to achieve a 7 phase shift. Tosteer a beam with a one-thousand pixel device, it would be necessary toactuate on the order of 1000 phase shifters spread out in a 1 mmaperture as shown in FIG. 6A. This is because, in this architecture,power is uniformly fed to all output antenna elements through a 3 dBsplitting tree and the thermal phase shifters are actively cohered toimplement in-plane beam steering over 1000 resolvable points. Actuatingthis many phase shifters would dissipate about 10 Watts.

Now consider the system of FIG. 1A with thermo-optic phase shifters tooperate the MZI switching matrix with N input ports to the planardielectric lens. The lens-based approach achieves N resolvable pointsin-plane by switching with an MZI tree switching matrix between N portsof a dielectric lens feed. The power requirements for the MZI treeswitching matrix in FIG. 6B scale like log₂ N as compared to N for thearchitecture in FIG. 6A because only MZIs associated with the desiredoptical signal path need to be activated (i.e., one MZI for each levelof the switching matrix hierarchy); the rest can be “off” and draw nopower. Consequently a lens-based device with 1000 resolvable points inplane dissipates 100 times less power for in-plane steering compared tothe conventional phase-shifter based approach shown in FIG. 6A. Thus,for lens-enabled LIDAR, the power budget is dominated by the opticalsignal generation, whereas for the phase-shifter architectures, itscales primarily with the feed size.

Most practical phase-shifter approaches require active feedback tomaintain beam coherence because thermal cross-talk causes changes in thepath length of neighboring waveguides. This means either making ameasurement of the relative phases on chip through lenses and detectorsor measuring the beam in free space through an IR camera to providefeedback. But a lens-based device does not to actively cohere thousandsof elements: it can use “binary”-like switching to route the light tothe appropriate port, which is a simpler control problem. This meansthat the beams emitted by the input ports to the lens can have arbitraryrelative phases. Lower power consumption additionally makes thermalfluctuations less severe.

Using a solid 1D grating reduces or eliminates grating lobes or highsidelobes that plague conventional optical phased arrays. This is at thecost of not being able to “constrain” the ray path to be in the forwarddirection and may result in having to use more material for the gratingcoupler, hence the triangle shape of the grating feed.

There is an alternate realization of this system, outlined below, whichdoes not use TO phase shifters and parallelizes the in-plane ports. Thisarchitecture parallelizes one scanning direction, as is commonly used inmost commercial LIDARs to increase scanning speed. This modification isnot possible with the conventional phase-shifter based approach.

Another advantage of a lens-based architecture is the ability to usealternate material systems. One reason for using Si for phased-arraydesigns is its large TO coefficient, which makes for lower power phaseshifters. However, the maximum IR power a single Si waveguide can carryis 50 mW, which significantly limits the LIDAR range. SiN has muchbetter properties in the IR and can take the order of 10 W through asingle waveguide. However, noting that the power required to operate thephase shifter goes like

${\frac{dT}{dn}\sigma},{{where}\mspace{14mu}\frac{dn}{dT}}$is the thermo-optic coefficient and σ is the conductivity, phaseshifters on the SiN platform may use at least three times more powerthan their Si equivalents. This would exacerbate the power budget andcontrol problems described above for any phase-shifter approach based onSiN. The lens-enabled design can still benefit from using SiN, andgreatly improve the potential range, because the feed power ispractically negligible.

No architecture is perfect, and there are several non-idealities whichcan alter the above story for our lens-based solution. The first is thenonuniform field of view of the device, which may cause problems forsome applications. Another concern is scaling the number of resolvablepoints to thousands of pixels in each scanning direction. Although it issimple to ray-trace a lens which would support up to a thousandresolvable points for in-plane scanning, implementing such a lens inpractice becomes more and more difficult because the requiredfabrication tolerances scale as 1/N. An additional concern is the impactof lens aberrations on the directivity degradation for the fullaperture. Although it was captured by ray-tracing, it was not rigorouslymodeled to determine the required tolerances and behavior for high Qgratings.

LIDARs with Lens-Enable Optical Beamformers

FIG. 7 shows a lidar system 700 that includes a lens-enabled,nonmechanical beam-forming system. The lidar system 100 includes atunable IR light source 780 that emits a tunable IR beam. An opticalpreamplifier 782 optically coupled to the tunable IR light source 780amplifies the tunable IR beam, which is coupled to a 1-to-128 MZI switchmatrix 710 via a 3 dB coupler 784 or coupler. The 3 dB coupler picks offa portion of the amplified beam for heterodyne detection of the receivedbeam with a detector 790. Signal processing electronics 792 coupled tothe detector 790 process the received signal.

The switch matrix 710 is fabricated on a SiN platform 708 that isintegrated with an InP platform 706 that supports a slab-coupled opticalwaveguide amplifier (SCOWA) array 712. This InP platrom is alsointegrated with another SiN platform 704 that includes a passivebeamforming chip 720 with both a planar dielectric lens and an outputcoupler. The lens may be a 20 nm or 40 nm thick PolySi lenses, andoutput coupling gratings may be 10 nm, 20 nm, or 40 nm thick. Thegratings support up to 300 resolvable points from wavelength tuning forthe 10 nm variants over a 100 nm bandwidth. Because of fabricationconstraints, the grating PolySi height may be the same as the lensheight. This can result in tradeoffs because thicker gratings had lowerquality factors, but thicker lenses have a better index contrast and cansupport more resolvable points. The switching matrix 710 is actuated byan off-chip digital controller 770.

In other examples, the system may be completely integrated. Forinstance, the tunable source, detectors, and electronics may beintegrated on the chip as the switch matrix, lens, and output coupler.Bringing all of these technologies together compactly, cheaply, androbustly yields a new sensor capable of supporting the next generationof autonomous machines.

FIG. 8 shows a lidar 800 with a tunable on-chip source 880, which can bemodulated with a microwave chirp. The chirped carrier travels through apreamplifier 882 and a directional coupler 884 and is split with a1-to-100 passive power splitter 810 to feed an array of 100 20 dBamplifiers 812 on a 10 μm pitch. Waveguides from these amplifiers 812are tapered to the edge of the beamforming aplanatic lens 820implemented with a SiN slab. These amplifiers 812 are turned on and offwith a digital controller 870 to the steer the output beam as explainedbelow. The lens 820 ends on a flat surface in front of a 1 mm², 2Dphotonic crystal (PhC) 830 that serves as the aperture and gratingcoupler. The reflected return from a target comes back through the sameaperture and is beat against a local oscillator (provided by the tunablesource 880) and undergoes balanced detection with a heterodyne detector890 coupled to signal processing electronics 892.

FIG. 9 shows an integrated optical beamforming system 900 that scalesthe basic design to add functionality for N independently controllablebeams. A seed from a tunable source 980 is split into 16 waveguides witha 1 by 16 power splitter 910 a. Each of these 16 power splitter outputsfeeds into its own preamplifier 982 and heterodyne detection unit 990,which are coupled to signal processing electronics 992. In turn, thesefeed 16 separate sections of the array with a power splitter 910 bcoupled to an array of 30 dB amplifiers 912 (here, 128 amplifiers)actuated by digital control electronics 970. A planar dielectric lens920 collimates the outputs of the amplifier array 912 for diffraction byan output coupler 930, such as a 1D grating or 2D photonic crystal.Here, 128 independent beams with scanning ranges limited to 128non-overlapping subsectors of the far field can be realized by turningon and off amplifiers connected to a given heterodyne detector.

The unit cells 800 and 900 shown in FIGS. 8 and 9 can be modified fortiling an M by N array as in FIG. 10A, which shows a unit cell 900modified to receive a local oscillator 1080 distributed among the tileswith waveguides. This seed 1080 is amplified with a preamplifier 1082and serves as the source for the tile. Before seeding the amplifiers982, the source phase is changed with a thermal phase shifter 1084.After the output lens 920, the neighboring subarrays are overlapped tosuppress the sidelobes of the tiled system. At the end of the grating930, the output light is sampled and undergoes balanced detection usinga balanced detector 1032 with the signal from a neighboring tile toprovide feedback to cohere the tiles.

FIG. 10B shows the tiling of the unit cells to form larger apertures.The cells are flipped on one side to expand the effective length of thePhC gratings.

Tunable Light Source and Preamplifier

As explained in greater detail below, the wavelength of the light sourcecontrols the out-of plane angle of the optical beam. Typical gratingantennas show steering at the rate of 0.1-0.2 degrees/nm. For instance,a light source with about 100 nm of tuning range provides a 12° to 16°field of regard. 2D photonic crystal gratings, discussed below, may haveenhanced steering rates. In addition, the laser provides seed power fordriving one or more optical amplifiers.

The power requirements for the laser source and optional preamplifiercan be determined by working through the signal chain for the completesystem. Consider a desired output of 500 mW/cm² for a system with 100input ports to the lens. This corresponds to 5 mW from a 1 mm² aperture.If there are 6 dB losses in the grating and lens, the input to the lensshould be about 20 mW. If the system includes an amplifier that provides20 dB gain (for this input port), the input power to the channel shouldbe 0.2 mW. To obtain 0.2 mW from a 1-to-128 splitter requires 20 mWignoring losses. This translates to 80 mW from the light source andpreamplifier, taking into account 6 dB losses from the splitter andcoupler. Assuming a nominal 10% efficiency, this preamplifier would need800 mW of electrical power for operation and 12 dB of gain given a 5 mWsource.

These specifications for an on-chip source are reasonable. A recent workdemonstrated a Vernier ring laser with 5.5 mW output power and a 41 nmtuning range. A thermal phase shifter allows tuning which can beadjusted on roughly 1 μs timescales, giving a sufficiently fastpoint-to-point sweep time for all realizations. This source may also bedirectly modulated with an RF chirp with a bandwidth of up to 9 GHzthrough plasma dispersion. RF modulation can also be implemented with anintegrated single sideband modulator.

1-to-N Optical Splitter

The system 100 in FIG. 1A includes a 1-to-N optical matrix 110. In theembodiments shown in FIGS. 8, 9 and 10A, the switch matrix is replacedby one or two passive 1-to-N splitter trees coupled to an array of Nsemiconductor amplifier switches (discussed below). The splitter tree iscreated using a binary tree of 50:50 splitters fabricated in SiN. Each50:50 splitter is an adiabatic 3 dB-coupler that is about 100-200 μmlong and about 10 μm wide. Thus a 1-to-128 splitter tree has 7 levels ofsplitting and is about 1 mm by 1 mm. A 1-to-1024 splitter tree has 10levels of splitting and is about 1.5 mm by 2 mm. Each splitter in thesplitter tree has a very low excess loss of about 0.1 dB. Thus, theentire splitter tree has a total excess loss of about 1 dB is expectedfor the entire tree. Because the splitting tree is before the amplifierbank and each amplifier is either off or on and saturated, the system isnot sensitive to minor variations in splitting ratio. This is incontrast to schemes in which splitters directly feed the grating couplerantennas. Adiabatic splitters are chosen to reduce or minimize backreflections and scattered light (which can be problematic in multimodeinterference (MMI) 3 dB splitters and Y-couplers) and to allow foruniform spitting over a wide optical bandwidth (>40 nm, which is notachieved using standard directional couplers).

For multiple beam and tiled realizations, a single laser source may feedinto an array of preamplifiers. Consider a tiled realization with 16tiles and a desired output of 500 mW from a 1 cm² aperture. If there are16 simultaneous beams, the preamplifier array should provide a 125 mWoutput keeping the same losses as above. For preamplifier with a 30 dBgain, the input power should be at least 0.125 mW and the output show beat least 30 mW from one of the 20 dB preamplifier units beforesplitting. The electrical power consumption for the preamplifiers formultiple beam and tiled realizations will be 4.8 W for a single chipassuming 10% efficiency.

The source and preamplifier devices may be created in InP and picked andplaced onto a passive SiN chip containing the splitters, lens, andgrating. For the single tile realization, there may be one combinedsource, preamplifier, and detection InP chip. For other realizations,one chip may have an array of preamplifiers and detectors and anotherchip may have the source.

Semiconductor Optical Amplifier (SOA) Switches

A passive splitter tree can be coupled to an array of SOA switches asshown in FIGS. 8, 9, and 10A, with each SOA switch amplifying orattenuating the input signal as a function of externally applied power(control signal). SOAs have fast (˜10 ns) switching speeds compared tothermal phase shifters (˜1 μs). SOAs having small-signal gain of 20 dB(e.g., in the single-tile realization in FIG. 8) and 30 dB (FIGS. 9 and10A) boost the seed laser output to provide the desired output power.For instance, the SOA output power may be 20 mW (FIG. 9), 125 mW (FIGS.9 and 10A), or any other suitable power level. SOAs can be integratedwith the SiN beamformer tiles using hybrid flip-chip integration. TheSOAs may be of any suitable type, including conventional orlow-confinement. Sample SOA specifications are shown in Table 1.

TABLE 1 Specifications for SOAs Gain Psat Length Realization (dB) (mW)Efficiency (cm) Approach Single Tile 20 20 10% <0.1 cm Conventional SOAST, MB 30 125 20% 0.32 cm High-confinement SCOWA Arrayed Tiles 30 12530% 0.32 cm High-confinement SCOWA

For example, each SOA may be implemented as the slab-coupled opticalwaveguide amplifier (SCOWA) developed by Lincoln Laboratory (LL). At1550-nm wavelength, a 1 cm long InP-based SCOWA having small-signal gainof 30 dB and saturation output power of 400 mW has been demonstrated. Byincreasing the SCOWA confinement factor appropriately, a SOA having 30dB gain and 125 mW output power should be realizable with a 0.32 cmlength. In addition to providing enough gain and output power for thisapplication, SCOWAs also have a very large transverse optical mode(e.g., about 5×5 μm), which increases the alignment tolerance when usingflip-chip integration to couple SOA and SiN chips. The flip-chipcoupling loss between a SCOWA and a SiP waveguide with the appropriatemode-size converter is about 0.5 dB to 1 dB.

For an array of conventional SOAs or SCOWAs, the minimum pitch is about10 μm to avoid optical coupling between neighboring devices. This smallpitch can be thermally managed as only one SCOWA is on at a time duringoperation. Therefore, arrays of 100 SCOWAs (single tile) and of 1000SCOWAs (other realizations) have footprints of 0.1×0.1 cm and 1×0.32 cm.respectively.

Since these SOAs amplify to 20 mW for the single-tile realization inFIG. 8, the design uses 200 mW electrical operating power assuming 10%efficiency. Given that the on-chip source uses 800 mW, the total powerrequirements for a feasible realization can be limited to 1 W. Thisgives a dissipated power density of approximately 17 W/cm². For thesingle-tile realization with multiple beams shown in FIG. 9, given that16 of the 125 mW amplifiers are on simultaneously and assuming 20%efficiency, the amplifier consumes 10 W. With 4.8 W for thepreamplifier/source unit, the dissipated power density comes to 6 W/cm²for the single-tile with multiple beam realization of FIG. 9 and lessthan 8 W/cm² for the arrayed tile realization in FIG. 10A.

Given an operating power of 1 W, and an output power of 5 mW, aconservative estimate for the wall plug efficiency for the realizationof FIG. 8 is about 0.5%.

For the realization of FIG. 9, the operating power may be about 15 W,including performance increases for the amplifier bank. With a 500 mWoptical output power, this puts the wall-plug efficiency at 3%.Improving to a 30% efficient amplifier bank yields 5% wall-plugefficiency for the realization of FIG. 10A.

The preamplifier architecture employed in the realizations of FIGS. 9and 10A makes it easier to increase the wall-plug efficiency. Thebenefit of using an array of preamplifiers mid-way through the splittingtree can be understood in the following way: it is most efficient to tumon and off individual sources at each input port of the beamforminglens. This means good solutions will reduce or minimize the power of thepreamplifier stage, increase or maximize the gain of the final amplifierbank, and turn off any unused amplifiers. In this case, placing an arrayof preamplifiers after the 1-to-16 splitter 910 a of FIGS. 9 and 10Areduces the power requirements for the preamplifier stage by a factor of16. Consequently, whereas the preamplifier and source uses four timesmore power than the final amplifier array for the realization of FIG. 8,for the realizations of FIGS. 9 and 10A it uses four times less,assuming equal efficiency of the components.

To create a 1 mm aperture, a conventional optical phased array designneeds on the order of 1000 thermal phase shifters. With an operatingpower of 20 mW/phase shifter, such a system would consume 20 W. This isan order of magnitude more power than the single-tile realization shownin FIG. 8. The favorable power scaling of the inventive approach extendsto other beamforming architectures.

Planar Dielectric Lens

The original microwave literature going back to 1946 explored the use oflenses for beam-steering applications. That literature was chieflyconcerned with mechanical displacement of the feed to obtain wide-angleand diffraction-limited beams. This specific approach was evenimplemented with MEMs and microlenses for small steering angles. Overtime, many mathematical techniques were developed to numericallycalculate the best lens shape to minimize aberrations which wouldotherwise quickly degrade the beam quality with increased steeringangle. Specifically lenses with wide-angle steering of −40 to +40degrees can be developed by numerically calculating a lens thatsatisfies a form of the Abbé sine condition.

Additional approaches to create lenses with similar wide angle rangesinclude bifocal and multi-focal lenses, which use additional degrees offreedom to create structures which have multiple perfect focal points inthe imaging surface. The Rotman lens is one such lens which utilizesdelay lines to create three focal points, one on-axis and two off-axis,for wide-angle steering. Graded index lenses such as the Luneburg lensallow for theoretically the widest angle steering possible by beingspherically symmetric. Beyond developing such a rich variety of lenses,the microwave literature also explored many techniques for optimallyfeeding the lenses, minimizing reflections, shaping the feed endaperture field patterns, and dealing with a myriad of other technicalproblems which may be relevant to our effort.

The planar dielectric lens can be implemented using any one of a varietyof designs. For example, it may be a dielectric slab lens with a singleperfect focal point in the imaging surface and numerically designed tosatisfy the Abbé sine condition. Fulfilling the Abbé sine conditiongives near diffraction limited performance to up to ±40 degrees. Fortiled realizations, alternate lenses, such as the Rotman lens or theLuneburg lens, may be employed to obtain up to 110 degrees or 180degrees of in-plane beam steering, respectively.

There are several approaches for implementing the lens in integratedphotonics, such as changing the height of the slab, patterning anadditional layer, doping, and varying the density of subwavelengthholes. These approaches have been used to implement GRIN lenses, such asthe Luneburg lens, on chip. For one implementation, a thin layer ofpolysilicon can be patterned on a silicon nitride slab. The high indexof polysilicon compared to silicon nitride creates a high effectiveindex contrast thereby increasing the focusing power of the lens.Adiabatically tapering the height of the lens can reduce the radiationlosses at the interface of the slab with the lens.

Coupler

To use the same antenna for transmit and receive, the antenna shouldcapture backward-propagating return power. Off-chip, an opticalcirculator would direct the backward propagating signal to the receiverwhile providing good transmit/receive (T/R) isolation. However, themagneto-optical materials used in such reciprocity breaking devices aredifficult to integrate on-chip. Instead, a simple adiabatic 3 dBsplitter sends half the backward propagating power to the receiver. Thelow loss and large optical bandwidths of such splitters should limit theperformance penalty to the 3 dB loss due to the splitting. This effectis especially small in the transmit direction as the splitter is locatedbefore the amplifier bank. Furthermore, since both the transmitter andreceiver ports are located on the same side of the device, decentisolation of the receiver from the transmitter is provided. Moresophisticated possibilities for transmitter-receiver isolation are alsopossible, including T/R switching or non-reciprocity from modulation.

Heterodyne Detection

The heterodyne detection shown in FIGS. 8, 9, and 10A can be performedwith two balanced InP detectors. In operation, these detectors recordthe heterodyne beat signal between the chirped source and the chirpedreturn. The use of balanced detectors allows for changes in theamplitude of the source and return to be decoupled from changes in theoffset frequency, giving a more robust measurement of round-trip time.

The outputs of the balanced detectors provide in-phase and quadrature(I/Q) signals with an intermediate frequency (IF) bandwidth determinedby the time-bandwidth product required from the transmitted linear-FMwaveform. The outputs of the I/Q detectors can be processed to createLIDAR imaging products.

Consider the simple example of a LIDAR on an autonomous vehicle. ThisLIDAR uses a 1 GHz linearly frequency-modulated (LFM) chirp over a 10 μsperiod (1500 m range gate) in a repeating sawtooth waveform.Stretched-pulse processing reduces the speed and power consumptionrequirements for the IF analog-to-digital converter (ADC) in the signalprocessing electronics on each receive channel. At zero time lag betweena target and the local oscillator (LO), the IF frequency is directcurrent (dc; 0 Hz). For a 0.2 μs (30 m) target-range displacement, theIF frequency is 20 MHz. An IF ADC with 50 Msamples/sec and a few bits ofdynamic range can easily detect the displaced target and accuratelydetermine its range with an uncertainty of ˜15 cm (0.5*c/1 GHz). Such acompact circuit can be implemented in a 65 nm complementarymetal-oxide-semiconductor (CMOS) process.

Overlapped Subarrays

Tiling creates a larger effective aperture as depicted in FIGS. 10A and10B. Even if neighboring tiles are properly cohered, there may be manynarrowly spaced sidelobes in the far-field because of the large distancebetween the center of each aperture. At the same time, this comb offar-field sidelobes may be modulated in magnitude by the far-field ofthe subarray pattern itself. Engineering the subarray pattern toresemble a sinc function in the near-field yields a box-like pattern inthe far-field that suppresses the sidelobes. It is possible to scan Nbeamwidths within this box, where N is the number of subarray elements.For example, in a tiled implementation, exciting 1 of 1,000 ports foreach tile and coherently phasing 10 neighboring tiles yields 10,000resolvable points.

To produce a sinc pattern in the near-field, there are severalapproaches to overlap and delay parts of the beam. One strategy is touse multiple waveguide layers to route light from the output grating ofone tile to another to form the larger subarray pattern. Thisarrangement can also work with a single layer of waveguides by utilizinglow-cross talk direct waveguide crossings. Another approach uses anarray of wedge-shaped microlenses or photonic crystals. Oneimplementation includes super-collimating photonic crystals to keep themain part of the beam going straight and defect waveguides to delay androute light to neighboring tiles.

Analysis of Optical Beam Steering with a Planar Dielectric Lens

The following analysis is intended to elucidate operation of an opticalbeam steering device with a planar dielectric lens. It is not intendedto limit the scope of the claims, nor is intended to wed such a deviceto particular mode or theory of operation.

Far-Field Angles

The aperture phase is determined by the initial ray directions, thegrating parameters, and the wavelength. Since we are propagating througha straight grating, the plane wave k generated by the lens feed systemwill be conserved, so k_(y,avg)=k_(y)·k_(x) on the other hand, will bemore complicated because it changes at each step of the grating.Assuming an initial in-plane angle of ϕ_(in) an index of the startingmedium n₁, an index of the steps n₂ and a step duty cycle d, we findthat k_(y) and k_(x,avg) are given by the following:

$\begin{matrix}{{k_{y} = {n_{1}k_{0}{\sin\left( \phi_{in} \right)}}}{k_{x,{avg}} = {{dn_{1}k_{0}\mspace{11mu}{\cos\left( \phi_{m} \right)}} + {\left( {1 - d} \right)n_{2}k_{0}\sqrt{\left( {1 - {\frac{n_{1}^{2}}{n_{2}^{2}}{\sin^{2}\left( \phi_{in} \right)}}} \right)}}}}} & (7)\end{matrix}$The effective indices for the grating n₁ and n₂ are also functions ofthe wavelength. To compute the emission angle of this aperture, weperform phase matching between these wavevectors and those of afree-space plane wave with {right arrow over (k)}=k₀[sin(θ₀)cos(ϕ₀),sin(θ₀)sin(ϕ₀), cos(θ₀)].

$\begin{matrix}{{{k_{0}{\sin\left( \theta_{0} \right)}{\sin\left( \phi_{0} \right)}} = {n_{1}k_{0}{\sin\left( \phi_{in} \right)}}}{{k_{0}{\sin\left( \theta_{0} \right)}{\cos\left( \phi_{0} \right)}} = {{k_{x,{avg}}\left( \phi_{in} \right)} - \frac{2\pi\; m}{\Lambda}}}} & (8)\end{matrix}$

Here we have subtracted a crystal momentum 2πm/Λ, which originates fromthe discrete and periodic sampling implemented by the scattering fromeach grating step. We can rearrange this to derive the followingexpressions for the far-field angles:

$\begin{matrix}{{\theta_{0} = \left( \frac{\sqrt{\left( {{k_{{avg},x}\left( \phi_{in} \right)} - \frac{2\pi\; m}{\Lambda}} \right)^{2} + \left( {n_{1}k_{0}\mspace{11mu}{\sin\left( \phi_{in} \right)}} \right)^{2}}}{k_{0}} \right)}{\phi_{0^{=}}\left( \frac{n_{1}k_{0}{\sin\left( \phi_{in} \right)}}{{k_{{avg},x}\left( \phi_{in} \right)} - \frac{2\pi\; m}{\Lambda}} \right)}} & (9)\end{matrix}$

We want to understand how the far-field angles depend on the in-planeangle ϕ_(in) and the wavelength λ. We can identify that ϕ₀ will besignificantly greater than ϕ_(in). This results from the gratingmomentum 2πm/Λ being subtracted from k_(avg,x) in the denominator. Thismeans that relatively small variations in the input angle will greatlychange the output in-plane angle ϕ₀, sweeping it across thefield-of-view. This feature ultimately allows us to use the lens in asmall angle, aplanatic regime.

As we sweep ϕ_(in) we also expect variations in θ₀. By examining (9), wesee that the argument of the arcsine term seems to increase with ϕ_(in)and ultimately exceed 1, confining the beam in-plane. We can derive thiscutoff condition (where θ₀=0) more precisely by takingk_(avg,x)(ϕ_(in))≈n_(eff)k₀ cos(ϕ_(in)), where n_(eff)=dn₁+(1−d)n₂. Eventhough n_(eff) is not truly constant, and its variations significantlyeffect the far-field angles, qualitatively this description holds. Wefind that the cutoff angle ϕ_(cut) satisfies:

$\begin{matrix}{1 = {\left\lbrack {{n_{eff}{\cos\left( \phi_{cut} \right)}} - \frac{\lambda}{\Lambda}} \right\rbrack^{2} + {n_{1}^{2}\left\lbrack {1 - {\cos\left( \phi_{cut} \right)}^{2}} \right\rbrack}}} & (10)\end{matrix}$

This can be easily rearranged into a quadratic equation and solved forϕ_(cut). To get more intuition into the behavior of this angle, weexamine the case of normal (or broadside) emission at ϕ_(in)=0 andapproximate n₁≈n_(eff). Working this out we find:

$\begin{matrix}{{\cos\left( \phi_{in} \right)} \approx {1 - \frac{1}{2n_{eff}^{2}}}} & (11)\end{matrix}$where tan(ϕ₀)≈−n_(eff)/2 gives the corresponding ϕ₀ at this point. Itmakes sense that the magnitude of the index will control the ϕ_(in)because it determines how rapidly k_(y) increases as we move off axis.Overall, we can envision how {right arrow over (k)} evolves as afunction of ϕ_(in): starting from emission normal to the surface, as weadjust ϕ_(in) away from 0, {right arrow over (k)} turns rapidly to oneside and falls into the plane.

We can further visualize this trajectory by rearranging (8). Takingu_(x,0)=sin(θ₀)cos(ϕ₀) and u_(y,0)=sin(θ₀) sin(ϕ₀), we can manipulate(8) to find:

$\begin{matrix}{{\left\lbrack \frac{u_{x,0} + \frac{m\;\lambda}{\Lambda}}{n_{eff}} \right\rbrack^{2} + \left\lbrack \frac{u_{y,0}}{n_{1}} \right\rbrack^{2}} = 1} & (12)\end{matrix}$This is an ellipse centered at

$\left\lbrack {{- \frac{m\;\lambda}{\Lambda}},0} \right\rbrack.$As ϕ_(in) is varied, the emission direction will traverse an arc of thisellipse in u_(x),u_(y) space. Tuning the wavelength λ will translatethis ellipse forward and backward in the u_(x) direction. The totalfield-of-view in u_(x),u_(y) space will have the form of a curved bandwhose thickness will be controlled by the total wavelength tuning range.We discuss the number of 3 dB overlapped beams we can fit inside thisfield-of-view below.

Far-Field Directivity

We can begin our derivation of the far-field pattern by noting that wecan completely specify the near-field amplitude to have the followingform:A(x,y)=exp(−qx)exp(ik ₀ u _(x0) x)exp(ik ₀ u _(y0) y)  (13)where u_(x0)=sin(θ₀)cos(ϕ₀) and u_(y0)=sin(θ₀)sin(ϕ₀).

We have implicitly assumed a rectangular beam profile along they-direction to simplify our calculations. In general we expect anadditional function ƒ(y,x) to modulate the amplitude of the patternaccording to the feed pattern, illumination position, and lens geometry.This derivation captures the most critical features of the far-fieldpattern and establishes an upper bound on the gain. In addition, theperformance of the aperture is largely determined by its phase behavior,so smearing the amplitude distribution relative to the ideal tends tolead to small changes.

The physical aperture we are integrating over is a parallelogram boundedby the following conditions:

$\begin{matrix}{0 \leq x \leq L} & (14) \\{{{- \frac{W}{2}} + {x\mspace{11mu}{\tan\left( \phi^{\prime} \right)}}} \leq y \leq {\frac{W}{2} + {x\mspace{11mu}{\tan\left( \phi^{\prime} \right)}}}} & (15)\end{matrix}$Here ϕ′ is equal to

$\left( \frac{n_{1}{\sin\left( \phi_{i\; n} \right)}k_{0}}{k_{x,{avg}}} \right) \approx \left( \frac{n_{1}{\sin\left( \phi_{i\; n} \right)}}{n_{eff}{\cos\left( \phi_{i\; n} \right)}} \right)$and is close in magnitude to ϕ_(in) from the previous section, but notidentical because of the refraction at the grating steps. To find thefar-field pattern we can compute the Fourier transform of this amplitudepattern over the domain:

$\begin{matrix}{{F\left( {{u_{x} - u_{x0}},{u_{y} - u_{y0}}} \right)} = {\int_{0}^{L}{{dx}{\int_{{- \frac{W}{2}} + {x\;{ta}\; n\;\phi^{\prime}}}^{\frac{W}{2} + {x\;{ta}\; n\;\phi^{\prime}}}{{dy}\;{\exp\left( {- {qx}} \right)}{\exp\left( {{- {{ik}_{0}\left( {u_{x} - u_{x\; 0}} \right)}}x} \right)}{\exp\left( {{- i}{k_{0}\left( {u_{y} - u_{y\; 0}} \right)}y} \right)}}}}}} & (16)\end{matrix}$Where u_(x)=sin(θ)cos(ϕ) and u_(y)=sin(θ)sin(ϕ), which are the directionangles. For convenience, from here we denote u_(x)−u_(x0) with Δu_(x)and u_(y)−u_(y0) with Δu_(y). We can evaluate these integrals easily tofind:

$\begin{matrix}{{F\left( {{\Delta u_{x}},{\Delta u_{y}}} \right)} = {\frac{\sin\left( {k_{0}\frac{W}{2}\Delta u_{y}} \right)}{k_{0}\Delta u_{y}}\frac{1 - {{\exp\left( {{- q}L} \right)}{\exp\left( {ik_{0}{L\left( {u_{x} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)}} \right)}}}{q + {i{k_{0}\left( {u_{x} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)}}}A}} & (17)\end{matrix}$The power of the far-field pattern is the magnitude of the field patternsquared, that is P=|F|². We use the power P below to compute thedirectivity of the far-field pattern with the following expression:

$\begin{matrix}{{D\left( {\theta,\phi} \right)} = \frac{P\left( {\theta,\phi} \right)}{\frac{1}{2\pi}{\int_{0}^{\pi}{d\;\theta\;\sin\;\theta{\int_{0}^{2\pi}{d\;\phi\;{P\left( {\theta,\phi} \right)}}}}}}} & (18)\end{matrix}$

The directivity gives the factor of the power emitted in a givendirection relative to an isotropic radiator. A well-designed directionalantenna tends to increase or maximize the peak gain, the directivity ofthe main lobe, and reduce or minimize the power into sidelobes, becausethese waste power and contribute to false detections. We will discussthe directivity more below concerning the range of the system and thenumber of resolvable points.

We can create a simpler expression by expanding the direction anglesabout the far-field peak at θ=θ₀ and ϕ=ϕ₀, we can also take the limitsof the integral to infinity. This creates negligible error in the caseof high-gain beams and ultimately allows many of these gain integrals tobe evaluated analytically:

$\begin{matrix}{{D\left( {{\Delta\theta},{\Delta\phi}} \right)} \approx \frac{P\left( {{\Delta\theta},{\Delta\phi}} \right)}{\frac{\sin\;\theta_{0}}{2\pi}{\int_{- \infty}^{\infty}{d\;{\Delta\theta}{\int_{- \infty}^{\infty}{d\;{\Delta\phi}\;{P\left( {{\Delta\theta},{\Delta\phi}} \right)}}}}}}} & (19)\end{matrix}$

It's illustrative to change coordinates of this expression from θ and ϕto Δu_(x) and Δu_(y). We can find:Δu _(x)=cos(θ₀)sin(ϕ₀)Δθ+sin(θ₀)cos(ϕ₀)Δϕ=u _(x) −u _(x,0)  (20)Δu _(y)=cos(θ₀)cos(ϕ₀)Δθ−sin(θ₀)sin(ϕ₀)Δϕ=u _(y) −u _(y,0)  (21)where we have taken ϕ=ϕ₀+Δϕ and θ=θ₀+Δθ. We can use these expressions tocalculate the following Jacobian, where we have changed variables from ϕand θ to Δϕ and Δθ:

$\begin{matrix}\begin{matrix}{{d\;{\Delta\theta}\; d\;{{\Delta\phi sin}\left( \theta_{0} \right)}} = {d\;\Delta\; u_{x}d\Delta u_{y}{\sin\left( \theta_{0} \right)}{\begin{matrix}\frac{\sin\left( \phi_{0} \right)}{\cos\left( \theta_{0} \right)} & \frac{\cos\left( \phi_{0} \right)}{\cos\left( \theta_{0} \right)} \\\frac{\cos\left( \phi_{0} \right)}{\sin\left( \theta_{0} \right)} & {- \frac{\sin\left( \phi_{0} \right)}{\sin\left( \theta_{0} \right)}}\end{matrix}}}} \\{= \frac{d\Delta u_{x}d\Delta u_{y}}{\cos\left( \theta_{0} \right)}}\end{matrix} & (22)\end{matrix}$Taken together, we can use these results to rewrite our expression forthe peak gain as a function of Δu_(x) and Δu_(y):

$\begin{matrix}{{D\left( {{\Delta u_{x}},{\Delta u_{y}}} \right)} = \frac{2\pi{\cos\left( \theta_{0} \right)}{P\left( {{\Delta u_{x}},{\Delta u_{y}}} \right)}}{\int_{- \infty}^{\infty}{d\;\Delta\; u_{x}{\int_{- \infty}^{\infty}{d\;\Delta\; u_{y}\;{P\left( {{\Delta\; u_{x}},{\Delta\; u_{y}}} \right)}}}}}} & (23)\end{matrix}$

This unsimplified expression can already tell us something veryuseful—that the peak gain of a given pattern is directly proportional tocos(θ₀). This result emerges because the far-field gain in general isproportional to the projected area. To first order, neglectingadditional aberrations and changes in the grating parameters, effectiveindices, reflections, and feed illumination, the peak gain fall-off as afunction of angle is just determined by the angle between the emissionvector and the z-axis. Another feature of this equation is that the peakshape is essentially independent of the center of the main lobe: toleading order the pattern just changes by the cos(θ₀) scale factor.

We can directly evaluate these integrals for our far-field pattern:

$\begin{matrix}{{\int_{- \infty}^{\infty}{d\Delta u_{x}{\int_{- \infty}^{\infty}{d\Delta u_{y}{P\left( {{\Delta u_{x}},{\Delta\; u_{y}}} \right)}}}}} = {\int_{- \infty}^{\infty}{d\;\Delta\; u_{x}{\int_{- \infty}^{\infty}{d\;\Delta\; u_{y}\frac{\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y}} \right)}}{1 + {\frac{k_{0}^{2}}{q^{2}}\left( {{\Delta u_{x}} + {{\tan(\phi)}\Delta\; u_{y}}} \right)^{2}}} \times \left( {1 - {2{\cos\left( {k_{0}{L\left( {{\Delta u_{x}} + {{\tan(\phi)}\Delta\; u_{y}}} \right)}} \right)}{\exp\left( {{- q}L} \right)}} + {\exp\left( {{- 2}qL} \right)}} \right)}}}}} & (24)\end{matrix}$We first start by performing a shear transformation on the integratingvariables given by: Δu_(x,s)=Δu_(x)+tan(ϕ)Δu_(y) and Δu_(y,s)=Δu_(y).With this transformation the integral now becomes separable:

$\begin{matrix}{{\int_{- \infty}^{\infty}{d\Delta u_{x}{\int_{- \infty}^{\infty}{d\Delta u_{x}{P\left( {{\Delta u_{x}},{\Delta\; u_{y}}} \right)}}}}} = {\int_{- \infty}^{\infty}{d\Delta u_{x,s}\frac{\left( {1 - {2{\cos\left( {k_{0}L\Delta u_{x,s}} \right)}{\exp\left( {{- q}L} \right)}} + {\exp\left( {{- 2}qL} \right)}} \right)}{1 + {\frac{k_{0}^{2}}{q^{2}}\Delta\; u_{x,s}^{2}}}{\int_{- \infty}^{\infty}{d\Delta u_{y,s}\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y,s}} \right)}}}}}} & (25)\end{matrix}$Note that the angle θ_(p) does not change the projected area of theaperture, since it just shears the emitting surface. Consequently weexpect it to completely drop out of the integral, which is indeed thecase. Next we remove the dimensions and break the integrals into partsand evaluate:

$\begin{matrix}{{\frac{2q}{{Wk}_{0}^{2}}{\int_{- \infty}^{\infty}{d{x\left( {\frac{\left( {1 + {\exp\left( {{- 2}qL} \right)}} \right)}{1 + x^{2}} - \frac{2{\cos\left( {qLx} \right)}{\exp\left( {{- q}L} \right)}}{1 + x^{2}}} \right)}{\int_{\infty}^{\infty}{{dy}\left( {}^{2}(y) \right)}}}}} = {\frac{2q\pi^{2}}{{Wk}_{0}^{2}}\left( {\left( {1 + {\exp\left( {{- 2}qL} \right)}} \right) - {2{\exp\left( {{- 2}qL} \right)}}} \right)}} & (26)\end{matrix}$Using these results, finally we can write the directivity as:

$\begin{matrix}{{D\left( {{\Delta u_{x}},{\Delta u_{y}}} \right)} = {\frac{{Wk}_{0}^{2}{\cos\left( \theta_{0} \right)}}{\pi{q\left( {1 - {\exp\left( {{- 2}qL} \right)}} \right)}}\frac{\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y}} \right)}}{1 + {\frac{k_{0}^{2}}{q^{2}}\left( {{\Delta u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)^{2}}} \times \left( {1 - {2{\cos\left( {k_{0}{L\left( {{\Delta u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)}} \right)}{\exp\left( {{- q}L} \right)}} + {\exp\left( {{- 2}qL} \right)}} \right)}} & (27)\end{matrix}$We can also expression the peak directivity as:

$\begin{matrix}{D_{{ma}\; x} = {{\cos\left( \theta_{0} \right)}\frac{{Wk}_{0}^{2}}{\pi q}\frac{1}{1 + \frac{k_{0}^{2}}{q^{2}}}\frac{\left( {1 - {\exp\left( {{- q}L} \right)}} \right)^{2}}{\left( {1 - {\exp\left( {{- 2}qL} \right)}} \right)}}} & (28)\end{matrix}$

To gain a little insight into how this function behaves, we can simplifyit for large and small L. For L<<1/q, we find:

$\begin{matrix}{{\lim\limits_{{qL}->0}{D\left( {{\Delta\; u_{x}},{\Delta\; u_{y}}} \right)}} = {\frac{{WLk}_{0}^{2}{\cos\left( \theta_{0} \right)}}{2\pi}\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y}} \right)}\sin\;{c^{2}\left( {\frac{L}{2}{k_{0}\left( {{\Delta u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)}} \right)}}} & (29)\end{matrix}$This is just the directivity from a sheared rectangular aperture oflength L and width W, note that the peak gain is

$\frac{{WLk}_{0}^{2}{\cos\left( \theta_{0} \right)}}{2\pi},$which is directly proportional to the projected area WLcos(θ₀). Takingthe opposite limit, we can find another useful simplification:

$\begin{matrix}{{\lim\limits_{{qL}->\infty}{D\left( {{\Delta\; u_{x}},{\Delta\; u_{y}}} \right)}} = {\frac{{Wk}_{0}^{2}{\cos\left( \theta_{0} \right)}}{\pi q}\frac{\sin\;{c^{2}\left( {\frac{W}{2}k_{0}\Delta u_{y}} \right)}}{1 + {\frac{k_{0}^{2}}{q^{2}}\left( {{\Delta u_{x}} + {{\tan\left( \phi^{\prime} \right)}\Delta\; u_{y}}} \right)^{2}}}}} & (30)\end{matrix}$

Here the peak directivity scales as

${\frac{W}{q}{\cos\left( \theta_{0} \right)}},{{where}\mspace{14mu}\frac{1}{q}}$becomes the effective length of the aperture. Even though theseexpressions are much simpler than the general one we derived, even ifthe aperture is several decay lengths long, the effect of the finitelength of the directivity is significant and properly modeling itrequires the full expression. An example of this is in computing thenumber of far-field resolvable points.

Number of Resolvable Points with Wavelength Tuning

Another property of our system is the number of far-field resolvablepoints. There are some relatively simple expressions we can derive whichwill tightly bound the number of resolvable points we can achieve in aparticular system as a function of the aperture parameters. We willfirst start with the number of resolvable points we can achieve throughwavelength tuning. Assuming normal incidence from the feed, thefar-field condition for the unit vector in the x-direction is just:

${{{n_{eff}k_{0}} - {\frac{2\pi}{\Lambda}m}} = {k_{0}u_{x}}},$where u_(x) is the unit vector of the wavevector in the x-direction,n_(eff) is the effective index of the grating at normal incidence, A isthe grating period, and m is the grating order.

We want to count the number of full-width half-maximums Δu_(FWHM), wecan fit inside a total tuning range of Δu_(range). Δu_(range) in thiscase is just given by

$\frac{m\Delta\lambda}{\Lambda},$where Δλ is the tuning wavelength, and is typically 50-100 nm forintegrated tunable sources. With this we can write a simple expressionfor the number of resolvable points with wavelength tuningN_(wavelength):

$\begin{matrix}{N_{wavelength} \approx {\frac{m\Delta\lambda}{\Lambda_{0}\Delta u_{FWHM}} + 1}} & (31)\end{matrix}$

In the case of a finite length grating, Δu_(FWHM) is computednumerically from the full directivity formula to give a precisecalculation of the number of resolvable points. However, in the case ofa long grating, we can determine exactly that

${\Delta u_{FWHM}} = {\frac{2q}{k_{0}}.}$Plugging this in gives the following relationship:

$\begin{matrix}{N_{wavelength} \approx {{\pi\frac{\Delta\lambda}{\lambda_{0}}\frac{m}{q\Lambda_{0}}} + 1}} & (32)\end{matrix}$Assuming that at λ₀, that the grating is emitting at normal incidence,and using our expression relating the decay length q to the gratingquality factor Q, we find that:

$\begin{matrix}{N_{wavelength} \approx {{Qn_{eff}\frac{\Delta\lambda}{\lambda_{0}}\frac{v_{g}}{c}} + 1}} & (33)\end{matrix}$

Number of Resolvable Points from in-Plane Steering

We assume that we can determine the number of resolvable points from thefield pattern at the lens aperture, as opposed to the pattern afterbeing emitted from the grating. In 1D, the directivity of a far-fieldpattern A(θ) is defined by

$\frac{A}{\frac{1}{\pi}{\int{{A(\theta)}d\theta}}}.$Assuming that the power is confined to a single lobe of angular widthΔθ, we can approximate D_(peak) as π/Δθ. Neglecting lens aberrations,the directivity can be written:

$\begin{matrix}{{D(\theta)} = {\frac{\pi}{\Delta\theta}{\cos(\theta)}}} & (34)\end{matrix}$

The steering range in this situation is limited by the minimumacceptable gain usable by the system. Typically RADARs are designed tohave a directivity fall-off of 3 dB or 0.5 at the edge of their usableFOV. This gives an effective steering range of 2π/3 radians.Conveniently approximating the beam-width to be constant, we find that:

$\begin{matrix}{{N_{i\; n\text{-}{plane}} \approx \frac{2D_{peak}}{3}} = \frac{2\pi W}{3\lambda}} & (35)\end{matrix}$where we have substituted in the peak directivity of a rectangularaperture of size W. This equation accurately reflects the scaling of thenumber of resolvable points a/λ when is between 10 and 40 or so. Beyondthis, the path error for off-axis scanning angles begins to become anappreciable fraction of the wavelength (since the error is directlyproportional to the lens size). The 3 dB scanning limit will be squeezedinwards as a/λ increases.

Abbé Sine Condition

If desired, we can shape a lens to satisfy the Abbé sine condition.Satisfying the Abbé sine condition eliminates Coma aberration on-axisand reduces it off axis in the regime where sin(ϕ)=ϕ. We briefly outlinethe procedure for generating a shaped lens given input parametersthickness T, focal length F, effective focal length F_(e), and index n.The inner surface of the lens is defined by r, θ, while the outersurface is defined by x, y. In this coordinate system, we satisfy theAbbé sine condition when y=F_(e) sin(θ). We can further relate r and θto x and y from the following expression calculated fromray-propagation:r+n√{square root over ((y−r sin(θ))²+(−r cos(θ))²)}−x=(n−1)T  (36)This can be written as a quadratic equation for x and solved. Once x issolved, r can be advanced by computing:

$\begin{matrix}{{\frac{dr}{d\theta} = \frac{nr{\sin\left( {\theta - \theta^{\prime}} \right)}}{{n{\cos\left( {\theta - \theta^{\prime}} \right)}} - 1}}{{Where}\text{:}}} & (37) \\{\theta^{\prime} = {\tan^{- 1}\left\lbrack \frac{\left( {F_{e} - r} \right){\sin(\theta)}}{x - {r{\cos(\theta)}}} \right\rbrack}} & (38)\end{matrix}$These equations can be solved iteratively to generate the entire lenssurface, beginning with θ=0 and r=F. Other methods can be used togenerate shaped lens surfaces, such as designing the aperture powerpattern based on the feed power pattern or forcing the lens to have twooff-axis focal points.

LIDAR Range

Generally, the minimum detectable received power P_(r,min) from a LIDARreturn determines the maximum range of the device. P_(r,min) isdetermined by the integration time and sensor architecture, which can bebased on frequency modulated continuous wave (FMCW) or pulsed directdetection type schemes. If a target has a cross section σ, the maximumrange we can observe that target is given by the standard RADARequation:

$\begin{matrix}{P_{r,{m\; i\; n}} = {\frac{{D\left( {\theta,\phi} \right)}^{2}\eta^{2}}{\left( {4\pi} \right)^{3}}\frac{\lambda^{2}}{R_{{ma}\; x}^{4}}\sigma\; P_{t}}} & (39)\end{matrix}$where D(θ,ϕ) is the directivity, θ is the device efficiency, R_(max) isthe maximum range, and P_(t) is the transmitter power. We see here thatthe primary determinant of the LIDAR performance beyond the detectionbackend are the antenna characteristics given by D(θ,ϕ) and η.

In the case that the beam spot from the LIDAR is contained completelywithin the target, which is a common application mode for LIDARs, we canderive an alternate constraint, which is more forgiving than thestandard RADAR range equation in terms of distance falloff:

$\begin{matrix}{P_{r,{m\; i\; n}} = {\frac{{D\left( {\theta,\phi} \right)}\eta}{\left( {4\pi} \right)^{2}}\frac{\lambda^{2}}{R_{{ma}\; x}^{2}}P_{t}}} & (40)\end{matrix}$

Overview of Numerical Methods and Verification

Because of structures are large and lack periodicity, full 3D FDTDsimulations were not possible. However, we were able to do smaller 2Dand 3D FDTD simulations of individual components to help verify thesystem performance.

First, we conducted simulations of the waveguides generated from therouting algorithms to verify that they were defined with enough points,were not too close, and satisfied minimum bend radius requirements.Unfortunately, having 3 dB spaced far-field spots results inwavelength-spaced ports in the focal surface. Although the waveguidescan be wavelength-spaced for short lengths without significant coupling,generally the feed geometry results in excessively high coupling betweenwaveguides. We fixed this problem by decimating the ports by a factor oftwo.

FIG. 11 shows 32 simulated 3 dB overlapped far-field beam patterns. Eachmain peak represents a single port excitation. Peaks to each side ofmain peak are sidelobes. These represent power radiated in unintendeddirections and may result in false detections. The dotted line indicates−3 dB. Port positions are designed to overlap far-field resolvable spotsby 3 dB. For a 2D aperture, this is done along parameterized curvebetween each spot in u_(x),u_(y) space.

FIGS. 12A-12D show simulations for design of chip-scale LIDAR. FIG. 12Ashows a simulation of a far-field beam pattern to extract phase center.FIG. 12B shows a simulation of a far-field beam pattern to extractgaussian beamwidth. FIG. 12C shows a 2D simulation of on-axis portexcitation of a lens feed. FIG. 12D shows a 2D simulation of off-axisport excitation of the lens feed.

The simulations of FIGS. 12A and 12B involve the interface between thewaveguide and the SiN slab. With these simulations, we visualized thecreation of an effective point dipole source when the waveguideterminated at the slab, and determined the far-field radiation patternand effective phase center. By fitting circles to the phase front, wewere able to extract a location for the phase center, which we show inFIG. 12A. The phase front was approximately 1 μm behind the interfacebetween the waveguide slab and the waveguide. The far-field power as afunction of angle was well described by a Gaussian with a beam-width of13.5°, as shown in FIG. 12B. The phase center and beam-width did notchange if the waveguide was incident at an angle on the interface: thephase center and beam-profile remained the same relative to theorientation of the waveguide. This feature was useful because it allowedus to angle the waveguides to reduce or minimize the spillover loss (theradiation that misses the lens) without having to be concerned about thebeam-width or phase center changing.

Another set of simulations we performed concerned the interface of thelens and the slab. One assumption, which is also a feature of otherworks on integrated planar lenses, is that we can describe the in-planepropagation in terms of the effective mode indices. We did severalcalculations of TE slab modes impacting 20 nm and 40 nm Si slab “steps”to verify this assertion and to quantify the radiation loss at theseinterfaces. We found that for a wide range of angles, the radiation losswas less than 5% in line with previous experimental results for incidentangles less than 40°.

We also performed effective 2D FDTD simulations of the waveguides, thelens feed, and the lens itself to verify that beam-steering workedproperly. We see this in FIGS. 12C and 12D. We confirmed the directivityderived from these simulations closely matched those produced byray-tracing. We also verified that the expected lens roughness forfabrication would not result in excessive gain degradation. 2D gratingsimulations of a 1D grating were used to extract the grating Q as afunction of wavelength. The emission angle was compared to thatpredicted from the average grating index and good agreement wasobtained. Additionally we modeled the photonic bandstructure using meepto confirm that our excitation was far from the Bragg band edge.

Finally, we extracted the grating Q as a function of angle andwavelength from meep calculations. We confirmed the on-axis performancematched that predicted from the FDTD simulations. Additionally weconfirmed that the Q did not change too much for off-axis propagation.Generally the behavior within ±20° was well-behaved, but beyond thatthere were large fluctuations. For the ray-tracing simulations, in theregime of interest, the grating Q could be considered constant, but ingeneral it was a complicated, rapidly varying function. Rigorouslymodeling Q as a function of angle accounts for the unexpectedly strongdependence in certain regimes.

Index Error

The effective index ratio n₂/n₁ of an experimental system is differentthan that used in ray-tracing simulations, because of finite fabricationtolerance, wavelength dispersion, temperature variation, etc. Ingeneral, an error in the index may cause the focal plane to shift bysome amount. For a parabolic lens, we find that the change is:

$\begin{matrix}{{\Delta f} = {\frac{f^{2}}{R}\Delta n}} & (41)\end{matrix}$

Since the depth of focus scales as λ, R:ƒ, and ƒ:λN, where N is thenumber of resolvable points, we have that our effective index tolerancescales inversely with the number of resolvable points that the imagingsystem supports:

$\begin{matrix}{{\Delta n} \approx \frac{1}{N}} & (42)\end{matrix}$Without any kind of external tunablity, meeting this constraint forlarge N becomes increasingly difficult. For more than 100 ports,wavelength dispersion over a 100 nm bandwidth already exceeds thisconstraint for a 40 nm thick lens. For proper operation of a device with100 ports at a single wavelength, there should be better than ±1 nm ofprecision in the layer heights, and better than 0.01 tolerance in thematerial index. Addressing these index tolerance issues enables scalingthe system to 1000s of resolvable points.

CONCLUSION

While various inventive embodiments have been described and illustratedherein, those of ordinary skill in the art will readily envision avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications isdeemed to be within the scope of the inventive embodiments describedherein. More generally, those skilled in the art will readily appreciatethat all parameters, dimensions, materials, and configurations describedherein are meant to be exemplary and that the actual parameters,dimensions, materials, and/or configurations will depend upon thespecific application or applications for which the inventive teachingsis/are used. Those skilled in the art will recognize, or be able toascertain using no more than routine experimentation, many equivalentsto the specific inventive embodiments described herein. It is,therefore, to be understood that the foregoing embodiments are presentedby way of example only and that, within the scope of the appended claimsand equivalents thereto, inventive embodiments may be practicedotherwise than as specifically described and claimed. Inventiveembodiments of the present disclosure are directed to each individualfeature, system, article, material, kit, and/or method described herein.In addition, any combination of two or more such features, systems,articles, materials, kits, and/or methods, if such features, systems,articles, materials, kits, and/or methods are not mutually inconsistent,is included within the inventive scope of the present disclosure.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of” or“exactly one of” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

The invention claimed is:
 1. An optical beam steering apparatuscomprising: a substrate; a plurality of waveguides formed on thesubstrate; a Luneburg lens, formed on the substrate in opticalcommunication with the plurality of waveguides, to direct light emittedby the plurality of waveguides in different directions; and an outputcoupler, formed on the substrate in optical communication with theLuneburg lens, to couple at least a portion of the light out of a planeof the Luneburg lens.
 2. The optical beam steering apparatus of claim 1,wherein the Luneburg lens has a field of view of about 180 degrees. 3.The optical beam steering apparatus of claim 1, wherein the Luneburglens is a planar lens having a graded index profile.
 4. The optical beamsteering apparatus of claim 1, wherein the Luneburg lens comprises alayer of polysilicon patterned on silicon nitride.
 5. The optical beamsteering apparatus of claim 1, wherein the output coupler is configuredto resolve at least 100 wavelengths.
 6. The optical beam steeringapparatus of claim 1, further comprising: a tunable light source, inoptical communication with the plurality of waveguides, to tune thewavelength of the light.
 7. The optical beam steering apparatus of claim6, further comprising: a network of optical switches, formed on thesubstrate in optical communication with the tunable light source and theplurality of waveguides, to guide the light from the tunable lightsource to a first waveguide in the plurality of waveguides when in afirst state and to guide the light emitted by the second waveguide inthe plurality of waveguides when in a second state.
 8. The optical beamsteering apparatus of claim 7, wherein the plurality of waveguidescomprises N waveguides and wherein up to log₂ N optical switches in thenetwork of optical switches are actuated to switch from the first stateto the second state.
 9. The optical beam steering apparatus of claim 7,further comprising: a plurality of optical amplifiers, formed on thesubstrate in optical communication with the network of optical switchesand the plurality of waveguides, to amplify the light emitted by thefirst waveguide and the light emitted by the second waveguide.
 10. Amethod of optical beam steering with a plurality of waveguides, aLuneburg lens, and an output coupler integrated with a substrate, themethod comprising: guiding light with the plurality of waveguides to theLuneburg lens; directing the light from the plurality of waveguides tothe output coupler with the Luneburg lens; and coupling at least aportion of the light out of the plane of the Luneburg lens with theoutput coupler.
 11. The method of claim 10, further comprising: tuning awavelength of the light so as to change an angle between the light andthe plane of the Luneburg lens.
 12. The method of claim 10, furthercomprising: guiding the light to the plurality of waveguides with anetwork of optical switches formed on the substrate.
 13. The method ofclaim 12, wherein the plurality of waveguides comprises N waveguides andfurther comprising: actuating up to log₂ N optical switches in thenetwork of optical switches to switch the light between waveguides inthe plurality of waveguides.
 14. The method of claim 10, furthercomprising: amplifying the light with an optical amplifier formed on thesubstrate in optical communication with the plurality of waveguides. 15.The method of claim 10, further comprising: receiving light via theoutput coupler; guiding the light via the output coupler to the Luneburglens; and coupling the light into a waveguide in the plurality ofwaveguides with the Luneburg lens.
 16. A lidar comprising: a tunablelight source to emit a beam of light; a plurality of waveguides; atleast one optical switch, in optical communication with the tunablelight source and the plurality of waveguides, to guide the beam of lightto a waveguide in the plurality of waveguides; a Luneburg lens, inoptical communication with the plurality of waveguides, to collimate thebeam of light emitted by the waveguide as a collimated beam propagatingin a plane of the Luneburg lens; and a periodic structure, in opticalcommunication with the Luneburg lens, to diffract at least a portion ofthe collimated beam at an angle with respect to the plane of theLuneburg lens.
 17. The lidar of claim 16, wherein the tunable lightsource is tunable over a bandwidth of at least 100 nm.
 18. The lidar ofclaim 16, wherein the Luneburg lens has a field of view of about 180degrees.
 19. The lidar of claim 16, wherein the Luneburg lens ispatterned in a slab waveguide coupled to the plurality of waveguides.20. The lidar of claim 16, wherein the Luneburg lens is a planar lenshaving a graded index profile.
 21. The optical beam steering apparatusof claim 1, wherein the output coupler comprises a curved grating. 22.The optical beam steering apparatus of claim 1, further comprising: aslab waveguide, formed on the substrate in optical communication withthe plurality of waveguides and the Luneburg lens, to guide the lightfrom the plurality of waveguides to the Luneburg lens.
 23. A method ofoptical beam steering with a planar Luneburg lens formed in a substratein optical communication with a waveguide array and a curved grating,the waveguide array comprising a first waveguide and a second waveguide,the method comprising: collimating, with the planar Luneburg, lightemitted by the first waveguide as a first collimated beam propagating ina first direction in a plane of the substrate; collimating, with theplanar Luneburg, light emitted by the second waveguide as a secondcollimated beam propagating in a second direction in the plane of thesubstrate different than the first direction; and coupling, with thecurved grating, at least a portion of the first collimated beam and thesecond collimated beam out of the plane of the substrate.
 24. The methodof claim 23, wherein collimating the light emitted by the firstwaveguide and collimating the light emitted by the second waveguideoccur simultaneously.
 25. The method of claim 23, further comprising:tuning a wavelength of the light emitted by the first waveguide and thelight emitted by the second waveguide.
 26. The method of claim 23,further comprising: coupling the light into the first waveguide via anoptical switch in a first state; actuating the optical switch from thefirst state to a second state; and coupling the light into the secondwaveguide via the optical switch in the second state.